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  • American Institute of Physics (AIP)  (49,884)
  • 1995-1999  (49,884)
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  • 1
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    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 4208-4226 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A simple technique employed almost three decades ago to manufacture partially solvable quantum many-body problems is revisited. [A quantum problem is "partially solvable" if (only) some of its eigenvalues and eigenfunctions can be exhibited]. The models thereby generated are characterized by Hamiltonians of normal form, i.e., standard kinetic plus momentum-independent potential energy; in most cases the latter features three-body, in addition to two-body and one-body, interactions. The setting refers to D-dimensional space; the examples focus on D=1, D=2, and D≥2, and include generalizations of, and additional results on, cases recently discussed in the literature, as well as new models. © 1999 American Institute of Physics.
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  • 2
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    Journal of Mathematical Physics 40 (1999), S. 1300-1316 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A mathematical framework for the completely positive semigroup coupling between classical and quantum systems is proposed. The coupling ensures a flow of information from the quantum system to the classical one and the influence of the classics on the dynamics of the quantum system in a dissipative way. The classical evolution on average is modified by the expectation value of some quantum operator. Examples of a classical particle moving along a geodesic line in a curved space interacting with the quantum system, and the coupling of a two state quantum system to all pure states, are discussed. © 1999 American Institute of Physics.
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  • 3
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    Journal of Mathematical Physics 40 (1999), S. 1359-1375 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We consider the Manev potential, given by the sum between the inverse and the inverse square of the distance, in an anisotropic space, i.e., such that the force acts differently in each direction. Using McGehee coordinates, we blow up the collision singularity, paste a collision manifold to the phase space, study the flow on and near the collision manifold, and find a positive-measure set of collision orbits. Besides frontal homothetic, frontal nonhomothetic, and spiraling collisions and ejections, we put into the evidence the surprising class of oscillatory collision and ejection orbits. Using the infinity manifold, we further tackle capture and escape solutions in the zero-energy case. By finding the connection orbits between equilibria and/or cycles at impact and at infinity, we describe a large class of capture-collision and ejection-escape solutions. © 1999 American Institute of Physics.
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  • 4
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    Journal of Mathematical Physics 40 (1999), S. 1163-1179 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We discuss a finite rectangular well of a depth λ2 as a perturbation for the infinite one with λ as a perturbation parameter. In particular, we consider a behavior of energy levels in the well as functions of complex λ. It is found that all the levels of the same parity are defined on infinitely sheeted Riemann surfaces whose topological structures are described in detail. These structures differ considerably from those found in models investigated earlier. It is shown that perturbation series for all the levels converge what is in a contrast with the known results of Bender and Wu. The last property is shown to hold also for the infinite rectangular well with the Dirac delta barrier as a perturbation considered earlier by Ushveridze. © 1999 American Institute of Physics.
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  • 5
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    Journal of Mathematical Physics 40 (1999), S. 1255-1267 
    ISSN: 1089-7658
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    Topics: Mathematics , Physics
    Notes: The Calabi–Yau and Joyce manifolds used in string and M-theory compactifications have no continuous groups of isometries, but they often have nontrivial discrete (actually finite) isometry groups. Discrete isometries of nonsimply connected Riemannian manifolds do not necessarily map spin structures into themselves, however; thus, inconsistencies are possible if a spin connection is used to construct the gauge vacuum. We consider this problem in detail and show how it may be avoided. © 1999 American Institute of Physics.
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  • 6
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    Journal of Mathematical Physics 40 (1999), S. 4255-4273 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We introduce and perform a systematic study of a new exactly solvable model of sphere interactions in quantum mechanics : the δ′ interaction, formally given by −Δ+αδ′(|x|−R), x∈R3, R〉0, α∈R. We also consider the cases of a δ′ plus a Coulomb interaction and finitely many δ′-sphere interactions with support on concentric spheres. For all these models, we provide basic properties and discuss the stationary scattering theory. We also briefly discuss the δ′-sphere interaction of the second type. © 1999 American Institute of Physics.
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  • 7
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    Journal of Mathematical Physics 40 (1999), S. 4290-4295 
    ISSN: 1089-7658
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    Topics: Mathematics , Physics
    Notes: It is widely believed that Bell has proved there can be no consistent local extension of the quantum formalism. Against this, Angelidis has presented a hidden variable theory which, he claims, makes precisely the same predictions as the quantum formalism and which also satisfies locality. In this note, we argue that Angelidis' theory does not live up to its inventor's claims. © 1999 American Institute of Physics.
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  • 8
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    Journal of Mathematical Physics 40 (1999), S. 4274-4289 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The quantum mechanics of a spin 〈fraction SHAPE="CASE"〉12 particle on a locally spatial constant curvature part of a (2+1)-space–time in the presence of a constant magnetic field of a magnetic monopole has been investigated. It has been shown that these two-dimensional Hamiltonians have the degeneracy group of SL(2,c), and para-supersymmetry of arbitrary order or shape invariance. Using this symmetry we have obtained its spectrum algebraically. The Dirac's quantization condition has been obtained from the representation theory. Also, it is shown that the presence of angular deficit suppresses both the degeneracy and shape invariance. © 1999 American Institute of Physics.
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  • 9
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    Journal of Mathematical Physics 40 (1999), S. 4296-4302 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A boundary value problem for a 2×2 S-matrix is solved by the method of projection matrices. The S-matrix is assumed to be diagnoalized by orthogonal matrix U(k2)=Ω(k2)/det Ω(k2), where matrix elements of Ω(k2) are polynomials. The equation det Ω(k2)=0 has roots of any order. The example is considered for the roots with order 2. © 1999 American Institute of Physics.
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  • 10
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    Journal of Mathematical Physics 40 (1999), S. 4303-4324 
    ISSN: 1089-7658
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    Topics: Mathematics , Physics
    Notes: We discuss the classical mechanics on the Grassmannian and the disc modeled on the ideal L(2,∞). We apply methods of geometric quantization to these systems. Their relation to a flat symplectic space is also discussed. © 1999 American Institute of Physics.
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  • 11
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    Journal of Mathematical Physics 40 (1999), S. 4325-4330 
    ISSN: 1089-7658
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    Topics: Mathematics , Physics
    Notes: The w∞ algebra is a higher-spin extension of the Virasoro algebra. In this paper, we construct the gauge theory of w symmetry in terms of its representations. © 1999 American Institute of Physics.
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  • 12
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    Journal of Mathematical Physics 40 (1999), S. 1549-1573 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Inönü–Wigner contractions from the rotation group O(n+1) to the Euclidean group E(n) are used to relate the separation of variables in Laplace–Beltrami operators on n-dimensional spheres and Euclidean spaces, respectively. In this article we consider all subgroup type coordinates corresponding to different chains of subgroups of O(n+1) and E(n), respectively. In particular, the contractions relate the graphical formalism of "trees" on spheres to the "clusters" on Euclidean spaces (introduced in this article). The contractions are considered analytically on several levels: the vector fields realizing the Lie algebras, the complete sets of commuting operators characterizing separable coordinate systems, the coordinate systems themselves and the separated eigenfunctions. © 1999 American Institute of Physics.
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  • 13
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    Journal of Mathematical Physics 40 (1999), S. 1595-1626 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We present and study a class of functions associated with the two-particle quantum relativistic Calogero–Moser system with elliptic interactions. The functions may be viewed as joint eigenfunctions of two independent commuting analytic difference operators, one of which is the defining quantum dynamics; The second one is obtained by interchanging the step size and the imaginary period. The functions depend on parameters that are dense in the natural parameter domain. In essence, they consist of products of Weierstrass σ-functions and plane waves. The zeros of the σ-functions satisfy a constraint system encoding both Schrödinger equations at once. © 1999 American Institute of Physics.
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  • 14
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    Journal of Mathematical Physics 40 (1999), S. 1114-1114 
    ISSN: 1089-7658
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    Topics: Mathematics , Physics
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  • 15
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    Journal of Mathematical Physics 40 (1999), S. 601-624 
    ISSN: 1089-7658
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    Topics: Mathematics , Physics
    Notes: For a quantum field in a thermal equilibrium state we discuss the group generated by time translations and the modular action associated with an algebra invariant under half-sided translations. The modular flows corresponding to the algebras of the forward light cone and a space-like wedge admit a simple geometric description in two-dimensional models that factorize in light-cone coordinates. At large distances from the domain boundary compared to the inverse temperature, the flow pattern is essentially the same as time translations, whereas the zero temperature results are approximately reproduced close to the edge of the wedge and the apex of the cone. For each domain there is also a one-parameter group with a positive generator, for which the thermal state is a ground state. Formally, this may be regarded as a certain converse of the Unruh effect. © 1999 American Institute of Physics.
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  • 16
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    Journal of Mathematical Physics 40 (1999), S. 635-673 
    ISSN: 1089-7658
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    Topics: Mathematics , Physics
    Notes: An iterative map of the unit disk in the complex plane is used to explore certain aspects of self-dual, four-dimensional gauge fields (quasi)periodic in the Euclidean time. These fields are characterized by two topological numbers and contain standard instantons and monopoles as different limits. The iterations do not correspond directly to a discretized time evolution of the gauge fields. They are implemented in an indirect fashion. First, (t,r,θ,φ) being the standard coordinates, the (r,t) half-plane is mapped on the unit disk in an appropriate way. This provides an (r,t) parametrization of Z0, the starting point of the iterations and makes the iterates increasingly complex functions of r and t. These are then incorporated as building blocks in the generating function of the fields. We explain in what sense and to what extent some remarkable features of our map (indicated in the title) are thus carried over into the continuous time development of the fields. Special features for quasiperiodicity are studied. Spinor solutions and propagators are discussed from the point of view of the mapping. Several possible generalizations are indicated. Some broader topics are also discussed. © 1999 American Institute of Physics.
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  • 17
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    Journal of Mathematical Physics 40 (1999), S. 4393-4399 
    ISSN: 1089-7658
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    Topics: Mathematics , Physics
    Notes: We derive a generalized Gel'fand–Levitan integral equation for two block Ablowitz–Kaup–Newell–Segur systems. This is possible if we suppose that the matrix coupling coefficients are invertible and come from simple zeroes of the determinant of the diagonal blocks of the scattering matrix. © 1999 American Institute of Physics.
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  • 18
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    Journal of Mathematical Physics 40 (1999), S. 4400-4418 
    ISSN: 1089-7658
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    Topics: Mathematics , Physics
    Notes: A general discussion is given of weak limits of classical dynamical systems depending on a parameter. The resulting maps are shown to be invertible if and only if they define a group of measure preserving point transformations. The irreversible case automatically leads to positive bistochastic maps and is characterized in terms of convergence properties of the corresponding automorphisms of the observable algebra. Necessary and sufficient conditions are given for the limit to define a time-independent Markov process. Two models are discussed, for a particle in a periodic potential, and for a particle interacting with fixed configurations of external obstacles. © 1999 American Institute of Physics.
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  • 19
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    Journal of Mathematical Physics 40 (1999), S. 821-829 
    ISSN: 1089-7658
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    Topics: Mathematics , Physics
    Notes: A classification of variationally trivial Lagrangians on T*M which are invariant under the Lie algebra of infinitesimal gauge transformations of the principal bundle π:M×U(1)→M, is given. A characterization of Lagrangian densities on T*M which are invariant under the Lie algebra of all infinitesimal automorphisms of M×U(1) is also obtained. © 1999 American Institute of Physics.
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  • 20
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    Journal of Mathematical Physics 40 (1999), S. 869-883 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In this paper we study an incompressible inviscid fluid when the initial vorticity is sharply concentrated in N disjoint regions. This problem has been well studied when a planar symmetry is present, i.e., the fluid moves in R2. In this case we know that, when the diameter σ of each region supporting the vorticity is very small, the time evolution of the fluid is quite well described by a dynamical system with finite degrees of freedom called the "point vortex model." In particular the connection between this model and the Euler equation has been proved rigorously as σ→0. In the present paper we discuss the "stability" of the point vortex model with respect to a particular small perturbation of the planar symmetry. More precisely we consider a fluid moving in R3 with a cylindrical symmetry without swirl in which each vortex is no longer a straight tube, but a vorticity ring. We prove that large annuli of radii r(approximate)σ−β for any β〉0 remain "localized" and hence we obtain the point vortex model as σ→0. © 1999 American Institute of Physics.
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  • 21
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    Journal of Mathematical Physics 40 (1999), S. 924-938 
    ISSN: 1089-7658
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    Topics: Mathematics , Physics
    Notes: Multidimensional gravity interacting with intersecting electric and magnetic p-branes is considered for fields depending on a single variable. Some general features of the system behavior are revealed without solving the field equations. Thus, essential asymptotic properties of isotropic cosmologies are indicated for different signs of spatial curvature; a no-hair-type theorem and a single-time theorem for black holes are proved (the latter makes sense in models with multiple time coordinates). The validity of the general observations is verified for a class of exact solutions known for the cases when certain vectors, built from the input parameters of the model, are either orthogonal in minisuperspace, or form mutually orthogonal subsystems. From the nonexistence of Lorentzian wormholes, a universal restriction is obtained, applicable to orthogonal or block-orthogonal subsystems of any p-brane system. © 1999 American Institute of Physics.
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  • 22
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    Journal of Mathematical Physics 40 (1999), S. 980-1010 
    ISSN: 1089-7658
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    Topics: Mathematics , Physics
    Notes: Computations of the strong field generation of gravitational waves by black hole processes produces waveforms that are dominated by quasinormal (QN) ringing, a damped oscillation characteristic of the black hole. We describe here the mathematical problem of quantifying the QN content of the waveforms generated. This is done in several steps: (i) We develop the mathematics of QN systems that are complete (in a sense to be defined) and show that there is a quantity, the "excitation coefficient," that appears to have the properties needed to quantify QN content. (ii) We show that incomplete systems can (at least sometimes) be converted to physically equivalent complete systems. Most notably, we give a rigorous proof of completeness for a specific modified model problem. (iii) We evaluate the excitation coefficient for the model problem, and demonstrate that the excitation coefficient is of limited utility. We finish by discussing the general question of quantification of QN excitations, and offer a few speculations about unavoidable differences between normal mode and QNM systems. © 1999 American Institute of Physics.
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  • 23
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    Journal of Mathematical Physics 40 (1999), S. 6145-6166 
    ISSN: 1089-7658
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    Topics: Mathematics , Physics
    Notes: The motivation for the present paper lies in the fact that the literature concerning the Coulomb wave functions FL(η,ρ) and GL(η,ρ) is a jungle in which it may be hard to find a safe way when one needs general formulas for the Coulomb wave functions with complex values of the variable ρ and the parameters L and η. For the Coulomb wave functions and certain linear combinations of these functions we discuss the connection with the Whittaker function, the Coulomb phase shift, Wronskians, reflection formulas (L→−L−1), integral representations, series expansions, circuital relations (ρ→ρe±iπ) and asymptotic formulas on a Riemann surface for the variable ρ. The parameters L and η are allowed to assume complex values. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6189-6208 
    ISSN: 1089-7658
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    Topics: Mathematics , Physics
    Notes: The osp(1,2)-covariant Lagrangian quantization of irreducible gauge theories is generalized to L-stage reducible theories. The dependence of the generating functional of Green's functions on the choice of gauge in the massive case is discussed and Ward identities related to osp(1,2) symmetry are given. Massive first-stage theories with closed gauge algebra are studied in detail. The generalization of the Chapline–Manton model and topological Yang–Mills theory to the case of massive fields are considered as examples. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6234-6244 
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    Notes: We establish the Schlieder and the Borchers property for thermal field theories. In addition, we provide some information on the commutation and localization properties of projection operators. © 1999 American Institute of Physics.
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  • 26
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    Journal of Mathematical Physics 40 (1999), S. 6254-6265 
    ISSN: 1089-7658
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    Notes: A simple method is proposed to construct the spectral zeta functions required for calculating the electromagnetic vacuum energy with boundary conditions given on a sphere or on an infinite cylinder. When calculating the Casimir energy in this approach no exact divergencies appear and no renormalization is needed. The starting point of the consideration is the representation of the zeta functions in terms of contour integral, further the uniform asymptotic expansion of the Bessel function is essentially used. After the analytic continuation, needed for calculating the Casimir energy, the zeta functions are presented as infinite series containing the Riemann zeta function with rapidly falling down terms. The spectral zeta functions are constructed exactly for a material ball and infinite cylinder placed in a uniform endless medium under the condition that the velocity of light does not change when crossing the interface. As a special case, perfectly conducting spherical and cylindrical shells are also considered in the same line. In this approach one succeeds, specifically, in justifying, in mathematically rigorous way, the appearance of the contribution to the Casimir energy for cylinder which is proportional to ln(2π). © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6302-6318 
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    Notes: The Wigner–Weyl quantum-to-classical correspondence rule is nonunique with respect to coordinate choice. This ambiguity can be exploited to improve the accuracy of semiclassical approximations. For instance, the well-known Langer modification was recently derived by applying a coordinate transformation to the radial Schrödinger equation prior to using the Wigner–Weyl rule—albeit only by presuming exact quantum solutions for all nonradial degrees of freedom [J. J. Morehead, J. Math. Phys. 36, 5431 (1995)]. In this paper, the full classical Hamiltonian is derived in all degrees of freedom, using a (hyper)spherical coordinate Wigner–Weyl correspondence with a Langer-like modification of polar angles. For central force Hamiltonians, the new result is radially equivalent to that of Langer, and to the standard Cartesian form. The new correspondence is superior with respect to all angular momentum operators however, in that the resultant semiclassical eigenvalues are exact—a desirable goal, evidently achieved here for the first time. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6353-6365 
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    Notes: Harmonic maps from S2 to CPN−1 are introduced to construct low-energy configurations of the SU(N) Skyrme model. We show that one of such maps gives an exact, topologically trivial, solution of the SU(3) model. We study various properties of these maps and show that, in general, their energies are only a little higher than the energies of the corresponding SU(2) embeddings. Moreover, we show that the baryon and energy densities of the SU(3) configurations with baryon number B=3−6 are more symmetrical than their SU(2) analogs, thus suggesting that there exist solutions of the model with these symmetries. We also show that any SU(2) solution embedded into the SU(4) Skyrme model becomes a topologically trivial solution of this model. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6473-6490 
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    Notes: In this work we develop a general procedure for constructing the recursion operators for nonlinear integrable equations admitting Lax representation. Several new examples are given. In particular, we find the recursion operators for some KdV-type systems of integrable equations. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6558-6576 
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    Notes: Multidimensional model describing the cosmological evolution and/or spherically symmetric configuration with n+1 Einstein spaces in the theory with several scalar fields and forms is considered. When electro-magnetic composite p-brane ansatz is adopted, n "internal" spaces are Ricci-flat, one space M0 has a nonzero curvature, and all p-branes do not "live" in M0, a class of exact solutions is obtained if certain block-orthogonality relations on p-brane vectors are imposed. A subclass of spherically symmetric solutions (containing nonextremal p-brane black holes) is considered. Post-Newtonian parameters are calculated. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6598-6608 
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    Notes: We estimate from above the set of knots, Ω(n,μ), generated by closure of the n-string 1+1- and 2+1-dimensional braids of irreducible length μ(μ(very-much-greater-than)1) in the limit n(very-much-greater-than)1. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6526-6557 
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    Notes: Binary constrained flows of soliton equations admitting 2×2 Lax matrices have 2N degrees of freedom, which is twice as many degrees of freedom than in the case of monoconstrained flows. By using the normal method, their Lax matrices directly give rise to first N pairs of canonical separated variables for their separation of variables. We propose a new method to introduce the other N pairs of canonical separated variables and additional separated equations. The Jacobi inversion problems for binary constrained flows are established. Finally, the factorization of soliton equations by two commuting binary constrained flows and the separability of binary constrained flows enable us to construct the Jacobi inversion problems for some soliton hierarchies. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6626-6637 
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    Notes: An algorithm is presented for the evaluation of the even moments of the end-to-end separation of a polymer chain. The only assumption made in the distribution function is that the energy of the chain depends only on the individual bond lengths of the chain. This assumption covers many of the commonly used distribution functions. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6689-6691 
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    Notes: We give a simple proof of the Ališauskas–Jucys triple sum formula for Wigner 9j-symbols. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6682-6688 
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    Notes: The computation of the index of the Hessian of the action functional in semi-Riemannian geometry at geodesics with two variable endpoints is reduced to the case of a fixed final endpoint. Using this observation, we give an elementary proof of the Morse index theorem for Riemannian geodesics with two variable endpoints, in the spirit of the original Morse proof. This approach reduces substantially the effort required in the proofs of the theorem given previously [Ann. Math. 73(1), 49–86 (1961); J. Diff. Geom 12, 567–581 (1977); Trans. Am. Math. Soc. 308(1), 341–348 (1988)]. Exactly the same argument works also in the case of timelike geodesics between two submanifolds of a Lorentzian manifold. For the extension to the lightlike Lorentzian case, just minor changes are required and one obtains easily a proof of the focal index theorem previously presented [J. Geom. Phys. 6(4), 657–670 (1989)]. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6692-6700 
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    Notes: We show that certain two-term transformation formulas between basic hypergeometric series can easily be described by means of invariance groups. For the transformations of nonterminating 3φ2 series, and those of terminating balanced 4φ3 series, these invariance groups are symmetric groups. For transformations of 2φ1 series the invariance group is the dihedral group of order 12. Transformations of terminating 3φ2 series are described by means of some subgroup of S6, and finally the invariance group of transformations of very-well-poised nonterminating 8φ7 series is shown to be isomorphic to the Weyl group of a root system of type D5. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6701-6706 
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    Notes: In this paper the concepts of almost Hermitian and Kählerian supermanifolds are introduced. Using the classical Newlander–Nirenberg theorm, it is shown that an almost complex structure J is integrable if it is compatible with ΔL, the supermanifold analog of the Levi-Civita connection. It is proved that an almost Hermitian supermanifold is Kählerian if and only if ΔL is compatible with J. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 5306-5317 
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    Notes: By using the field–antifield formalism, we show that the method of Batalin, Fradkin, Fradkina, and Tyutin (BFFT) to convert Hamiltonian systems submitted to second class constraints introduces compensating fields which do not belong to the BRST cohomology at ghost number one. This assures that the gauge symmetries which arise from the BFFT procedure are not obstructed at quantum level. An example where massive electrodynamics is coupled to chiral fermions is considered. We solve the quantum master equation for the model and show that the respective counterterm has a decisive role in extracting anomalous expectation values associated with the divergence of the Noether chiral current. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 5318-5340 
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    Notes: Canonical quantization of a Chern–Simons gauge field minimally coupled to a spinor field is studied as an indefinite metric quantum field theory in the usual covariant gauges, by using a lattice cutoff. For this model, we show that positivity for the indefinite metric, Gauss' Law, gauge invariance and Osterwalder–Schrader positivity for a self-adjoint Hamiltonian are equivalent. In addition, the path-space formula for the Osterwalder–Schrader semigroup is constructed in terms of a Euclidean scalar, massive, Gaussian random field. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 5440-5455 
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    Notes: Recently Jarvis has proved a correspondence between SU(N) monopoles and rational maps of the Riemann sphere into flag manifolds. Furthermore, he has outlined a construction to obtain the monopole fields from the rational map. In this paper we examine this construction in some detail and provide explicit examples for spherically symmetric SU(N) monopoles with various symmetry breakings. In particular we show how to obtain these monopoles from harmonic maps into complex projective spaces. The approach extends in a natural way to monopoles in hyperbolic space and we use it to construct new spherically symmetric SU(N) hyperbolic monopoles. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 5456-5472 
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    Notes: We formulate an adiabatic theorem adapted to models that present an instantaneous eigenvalue experiencing an infinite number of crossings with the rest of the spectrum. We give an upper bound on the leading correction terms with respect to the adiabatic limit. The result requires only differentiability of the considered projector, and some geometric hypothesis on the local behavior of the eigenvalues at the crossings. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 5749-5786 
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    Notes: We show that certain partial differential equations associated to nonisospectral scattering problems in 2+1 dimensions provide a key to associated integrable hierarchies of both ordinary and partial differential equations. This is illustrated using (an extension of) a known second-order and two new third-order nonisospectral scattering problems. These scattering problems allow us to derive new hierarchies of integrable partial differential equations, in both 1+1 and 2+1 dimensions, together with their underlying linear problems (isospectral and nonisospectral); and also new hierarchies of integrable ordinary differential equations, again with their underlying linear problems. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 5835-5848 
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    Notes: In an open time-convex region Λ of a strongly causal Lorentzian manifold (M,g), we consider an event p and a timelike, injective curve γ. We look for geodesics connecting p and γ in Λ and satisfying the conservation law g(z)[z(overdot),z(overdot)]=−E for a fixed E〉0. It is already known that such geodesics are the stationary points of the arrival time functional τ. Our main result is to prove the existence of a decreasing flow for τ, by means of a shortening procedure. This makes possible to apply to τ global variational methods obtaining existence and multiplicity results (using the Ljusternik–Schnirelmann category theory) and also to develop a Morse theory. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 5593-5640 
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    Notes: For the description of space–time fermions, Dirac–Kähler fields (inhomogeneous differential forms) provide an interesting alternative to the Dirac spinor fields. In this paper we develop a similar concept within the symplectic geometry of phase spaces. Rather than on space–time, symplectic Dirac–Kähler fields can be defined on the classical phase space of any Hamiltonian system. They are equivalent to an infinite family of metaplectic spinor fields, i.e., spinors of Sp(2N), in the same way an ordinary Dirac–Kähler field is equivalent to a (finite) multiplet of Dirac spinors. The results are interpreted in the framework of the gauge theory formulation of quantum mechanics which was proposed recently. An intriguing analogy is found between the lattice fermion problem (species doubling) and the problem of quantization in general. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3791-3826 
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    Notes: We construct a free field realization of vertex operators of the dilute AL models along with the Felder complex. For L=3, we also study an E8 structure in terms of the deformed Virasoro currents. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3827-3842 
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    Notes: We present the covariant generalization of the Dirac equation in a nonfactorizable metric and give the corresponding exact solutions in terms of special functions as well as the explicit form of the spinor solution. Then we treat the particular case of the Weyl equation for the neutrinos. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3876-3880 
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    Notes: Let (l−k2)u=−u″+q(x)u−k2u=δ(x), x∈R, ∂u/∂|x|−iku→0, |x|→∞. Assume that the potential q(x) is real valued and compactly supported: q(x)=q(x), q(x)=0 for |x|≥1, ∫−11|q|dx〈∞, and that q(x) produces no bound states. Let u(−1,k) and u(1,k), ∀k〉0 be the data. It is shown that under the above assumptions these data determine q(x) uniquely. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 5883-5889 
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    Notes: A K slicing of the Reissner–Nordstrom (RN) spacetime is shown to provide a complete foliation of the region up to the inner horizon, and the horizon to correspond to the limit as K→±∞. The implications of this foliation in the context of York time are discussed. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 5905-5920 
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    Notes: This paper extends the investigation of the invariants of the Riemann tensor to include the invariants that are of odd degree in the trace-free Ricci tensor. It is shown that these invariants can be expressed in terms of 15 such invariants that are irreducible. As a consequence, it is possible to write down a complete set of invariants of the Riemann tensor. Several syzygies for these invariants have been found and examples of these are given. These syzygies suggest there may be several new syzygies of invariants with even degree in the trace-free Ricci tensor. A large number of these have also been found and are discussed in the paper. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 5890-5904 
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    Notes: For the first time, exact solutions for the full self-consistent Einstein–Yang–Mills–Dirac systems with gauge groups SU(2) and SU(4) in Robertson–Walker space–time R×S3 are presented, which are symmetric under the action of the group SO(4) of spatial rotations. The approach is based on the dimensional reduction method for gauge, spinor, and gravitational fields and relates symmetric solutions in EYMD theory to certain solutions of an effective dynamic system. The solutions are interpreted as solutions of the cosmological type with an oscillating Yang–Mills field, which exchanges energy with a spinor field. The explicit form of the solution for the spinor field shows that its energy changes the sign in the process of the evolution of the Yang–Mills field from one vacuum to another. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 5939-5955 
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    Notes: The explicit denominator (normalization) function of the canonical tensor operators of the quantum algebra uq(3), corresponding to the maximal null space case is derived ab initio in terms of double basic hypergeometric series, which cannot be obtained as any q-extension of the SU(3) denominator polynomial Gb″1(Δ,x) in terms of multiple (double or triple) balanced hypergeometric series, introduced by Biedenharn, Louck, and their collaborators (although their q=1 versions are shown being equivalent). The corresponding orthonormal seed isoscalar factors of the coupling (Wigner–Clebsch–Gordan) coefficients of uq(3) and SU(3) with multiple irreducible representations are presented. Conjectured expression of the q-polynomials [which ratios appear in the uq(3) and (new) SU(3) denominator functions for an arbitrary value of the canonical multiplicity label t of the repeating irreducible representations] in terms of multiple partition dependent q-series (extension of the maximal and minimal null space versions) is presented and considered. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 5956-5971 
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    Notes: We derive Galilean wavelets, by which we mean coherent states of the affine Galilei group, that is, the Galilei group extended by independent space and time dilations. The construction follows a general method based on square integrable group representations, possibly modulo a subgroup, i.e., on a homogeneous space of the underlying group. We also examine the restriction to the Schrödinger subgroup, which contains only dilations that leave invariant the Schrödinger and the heat equations. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6033-6043 
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    Notes: We make use of the properties of product integrals to obtain a surface product integral representation for the Wilson loop operator. The result can be interpreted as the non-Abelian version of Stokes' theorem. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6087-6109 
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    Notes: We describe the construction of some finite dimensional nonunitary representations of E(2), the Lie group of Euclidean transformations in the plane. Some properties of these representations are also discussed, with emphasis on indecomposable representations. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6071-6086 
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    Notes: A four-field lattice furnished by Blaszak and Marciniak is transformed into bilinear form upon introducing two auxiliary independent variables. A Bäcklund transformation in bilinear form is found for the lattice and the corresponding nonlinear superposition formula is rigorously established. As a consequence, soliton solutions to the lattice are derived. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6052-6070 
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    Notes: From the bicovariant first-order differential calculus on inhomogeneous Hopf algebra B we construct the set of right-invariant Maurer–Cartan one-forms considered as a right-invariant basis of a bicovariant B-bimodule over which we develop the Woronowicz general theory of differential calculus on quantum groups. In this formalism, we introduce suitable functionals on B which control the inhomogeneous commutation rules. In particular, we find that the homogeneous part of commutation rules between the translations and those between the generators of the homogeneous part of B and translations are controlled by different R-matrices satisfying characteristic equations. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 6126-6127 
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    Notes: © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 4807-4812 
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    Notes: The Landau–Lifshitz equation is derived as a first-order iteration of the Lorentz–Dirac equation for the charged particle. In those cases with null electromagnetic field's gradient, the Landau-Lifshitz gives the so named Herrera equation. A general method for the solution of the latter equation is presented and applied to the motion of a particle in a uniform electromagnetic field. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 4903-4910 
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    Notes: It is shown that the trajectories of a charged particle in a static magnetic field and a velocity-independent potential in a three-dimensional space are (the projection of) the geodesics of a suitably defined metric in a four-dimensional space. It is shown that each one-parameter group of isometries of the original configuration space that leaves the magnetic field invariant gives rise to a one-parameter group of isometries of the metric defined on the four-dimensional space and, hence, to a constant of the motion. It is also shown, similarly, that the Schrödinger equation for a charged particle in a static magnetic field is equivalent to the Schrödinger equation for a free particle in the four-dimensional space mentioned above. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 4911-4943 
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    Notes: Euclidean and Minkowski four-space uniqueness theorems are derived which yield a new perspective of classical four-vector fields. The Euclidean four-space uniqueness theorem is based on a Euclidean four-vector identity which is analogous to an identity used in Helmholtz's theorem on the uniqueness of three-vector fields. A Minkowski space identity and uniqueness theorem can be formulated from first principles and the space components of this identity turn out to reduce to the three-vector Helmholtz's identity in a static Newtonian limit. A further result is a uniqueness theorem for scalar fields based on an identity which is proved to be a static Newtonian limit of the zeroth or scalar component of the Minkowski space extension of the Helmholtz identity. Last, the three-vector Helmholtz identity and uniqueness theorem and their four-space extensions to Minkowski space are generalized to mass damped fields. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 4028-4034 
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    Notes: In this paper a theorem is derived in order to provide a wide sufficient condition for an orthogonally transitive cylindrical space–time to be singularity free. The applicability of the theorem is tested on examples provided by the literature that are known to have regular curvature invariants. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 4713-4781 
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    Notes: We propose the (BRST-invariant) quantum open superstring field theory in the "B0-gauge," based on Neveu–Schwarz (NS) strings in 1 picture and Ramond (R) strings in 〈fraction SHAPE="CASE"〉12 picture. We give the propagators of these open NS and R superstrings. In order to obtain the BRST-invariant interaction terms among these superstrings, we modify the interaction terms among three superstrings (i.e., among NS–NS–NS and R–R–NS) by subtracting the infinite number of counter terms, each of which involves interaction terms among "more than four superstrings." The modified action can be obtained successively, so that resulting amplitudes in g-loops should become BRST invariant. Thus obtained amplitudes are referred to as the "amputated scatts," with the help of which the physical scattering amplitudes can be expressed. These physical scattering amplitudes among NB bosonic (NF fermionic) particles are calculated by using the analytic inlint gluing operator (which has already been proposed and used in the quantum bosonic string field theory "in the B0=0 gauge"). © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 4828-4858 
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    Notes: We derive transport equations from a general class of equations of form iut=H(X,D)u+V(X,D)u where H(X,D) and V(X,D) are pseudodifferential operators (Weyl operator) with symbols H(x,k) and V(x,k), where H(x,k) being polynomial in k and smooth in x,V(x,k) is a mean zero random function and is stationary in space variable. We also consider system of equations in the above form. Such equations cover many of the equations that arise in wave propagations, such as those considered in a paper by Ryzhik, Papanicolaou, and Keller [Wave Motion 24, 327–370 (1996)]. Our results generalize those by Ryzhik, Papanicolau, and Keller. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 188-209 
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    Notes: Integrable systems in low dimensions, constructed through the symmetry reduction method, are studied using phase portrait and variable separation techniques. In particular, invariant quantities and explicit periodic solutions are determined. Widely applied models in Physics are shown to appear as particular cases of the method. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 156-168 
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    Notes: The stochastic functional method is applied to plane-wave scattering from a random cylindrical surface, whereupon the Dirichlet boundary condition is rigorously imposed. Analytical results, accurate to second and fourth order in surface roughness, are obtained for the coefficients of the Wiener–Hermite expansion of the secondary scattered wave field. The validity of approximate solutions is numerically investigated by means of the boundary condition criterion and of the energy consistency criterion. The former, which is introduced herein, states that any approximate solution should be in conformity with the boundary condition, whereas the latter pertains to the energy conservation law. The numerical investigation indicates that the rigorous application of the stochastic functional method yields more accurate results in terms of both criteria than did previous treatments of the problem under consideration. Moreover, it is suggested that applicability limits should be set through the mean boundary condition criterion instead of the energy consistency criterion; the latter may lead to underestimating deficiencies of the approximate solution under test. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 236-247 
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    Notes: The classical n-dimensional Calogero–Moser system is a maximally superintegrable system endowed with a rich variety of symmetries and constants of motion. In the first part of the article some properties related with the existence of several families of constants of motion are analyzed. In the second part, the master symmetries and the time-dependent symmetries of this system are studied. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 256-278 
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    Notes: An ideal fluid whose internal energy depends on density, density gradient, and entropy is considered. Dynamic equations are integrated, and a description in terms of hydrodynamic (Clebsch) potentials occurs. All essential information on the fluid flow (including initial and boundary conditions) appears to be carried by the dynamic equations for hydrodynamic potentials. Information on initial values of the fluid flow is carried by arbitrary integration functions. Initial and boundary conditions for potentials contain only nonessential information concerning the fluid particle labeling. It is shown that the description in terms of n-component complex wave function is a kind of such description in terms of hydrodynamic potentials. Spin determined by the irreducible number nm of the wave function components appears to be an attribute of the fluid flow. Classification of fluid flows by the spin appears to be connected with invariant subspaces of the relabeling group. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 4072-4083 
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    Notes: A subclass of a recently discovered class of solutions in multidimensional gravity with intersecting p-branes related to Lie algebras and governed by a set of harmonic functions is considered. This subclass in case of three Euclidean p-branes (one electric and two magnetic) contains a cosmological solution to D=11 supergravity related to hyperbolic Kac–Moody algebra F3 (of rank 3). This solution describes the non-Kasner power-law inflation. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 35-48 
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    Notes: In this work we describe a general class of dispersive N-body Hamiltonians for which we prove the Hunziker, van Winter, and Zislin (HWZ) theorem and a Mourre estimate outside a closed and countable set of energies called thresholds. As a consequence of the Mourre estimate we prove a strong form of the limiting absorption principle, which implies the absence of a singular continuous spectrum and gives criteria of local smoothness. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 4099-4105 
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    Notes: It is shown that there exists a one-to-one correspondence between the real solutions of the Einstein vacuum field equations linearized about the Minkowski metric and the (complex) metric perturbations whose curvature to first order in the metric perturbation is self-dual. It is also shown that the self-duality condition of the curvature to first order in the metric perturbation is equivalent to a set of first-order equations for the metric perturbation whose solution is given by a scalar potential that obeys the wave equation. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 4089-4098 
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    Notes: Prepotentials in N=2 supersymmetric Yang–Mills theories are known to obey nonlinear partial differential equations called Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations. In this paper, the prepotentials at the one-instanton level in N=2 supersymmetric SU(4) Yang–Mills theory are studied from the standpoint of WDVV equations. Especially, it is shown that the one-instanton prepotentials are obtained from WDVV equations by assuming the perturbative prepotential and by using the scaling relation as a subsidiary condition but are determined without introducing the Seiberg–Witten curve. In this way, various one-instanton prepotentials which satisfy both WDVV equations and the scaling relation can be derived, but it turns out that among them there exist one-instanton prepotentials which coincide with the instanton calculus. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 4106-4118 
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    Notes: The singular continuous spectrum of the Liouville operator of quantum statistical physics is, in general, properly included in the difference of the spectral values of the singular continuous spectrum of the associated Hamiltonian. The absolutely continuous spectrum of the Liouvillian may arise from a purely singular continuous Hamiltonian. We provide the correct formulas for the spectrum of the Liouville operator and show that the decaying states of the singular continuous subspace of the Hamiltonian do not necessarily contribute to the absolutely continuous subspace of the Liouvillian. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 4119-4133 
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    Notes: The global additive and multiplicative properties of the Laplacian on j-forms and related zeta functions are analyzed. The explicit form of zeta functions on a product of closed oriented hyperbolic manifolds Γ(backward-slash)Hd and of the multiplicative anomaly are derived. We also calculate in an explicit form the analytic torsion associated with a connected sum of such manifolds.© 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 4177-4178 
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    Journal of Mathematical Physics 40 (1999), S. 3327-3340 
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    Notes: It is argued that geometric algebra, in the form of the multiparticle spacetime algebra, is well suited to the study of multiparticle quantum theory, with advantages over conventional techniques both in ease of calculation and in providing an intuitive geometric understanding of the results. This is illustrated by comparing the geometric algebra approach for a system of two spin-1/2 particles with the nonrelativistic approach of Holland [Phys. Rep. 169, 294 (1988)]. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3311-3326 
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    Notes: We outline a method based on successive canonical transformations which yields a product expansion for the evolution operator of a general (possibly non-Hermitian) Hamiltonian. For a class of such Hamiltonians this expansion involves a finite number of terms, and our method gives the exact solution of the corresponding time-dependent Schrödinger equation. We apply this method to study the dynamics of a general nondegenerate two-level quantum system, a time-dependent classical harmonic oscillator, and a degenerate system consisting of a spin 1 particle interacting with a time-dependent electric field E(vector)(t) through the Stark Hamiltonian H=λ(J(vector)⋅E(vector))2. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3409-3430 
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    Notes: Under a constraint between the potentials and the eigenfunctions, the 3×3 AKNS matrix spectral problem and its adjoint spectral problem associated with the three-wave interaction equations are nonlinearized so as to be a new finite-dimensional Hamiltonian system. A general scheme for generating involutive systems of conserved integrals and their two new generators are proposed, by which the finite-dimensional Hamiltonian system is further proved to be completely integrable in the Liouville sense. Moreover, the involutive solutions of the three-wave interaction equations are given. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3431-3438 
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    Notes: Let (P,π,B,G) be a G-principal fiber bundle. The action of G on the cotangent bundle T*P is free and Hamiltonian. By Liberman and Marle [Symplectic Geometry and Analytical Mechanics (Reidel, Dortrecht, 1987)] and Marsden and Ratiu [Lett. Math. Phys. 11, 161 (1981)] the quotient space T*P/G is a Poisson manifold. We will determine the Poisson bracket on the reduced Poisson manifold T*P/G, and its symplectic leaves. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3439-3445 
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    Notes: Cauchy analysis of the linearized field equations of the skew sector of the massive nonsymmetric gravitational theory shows that small perturbations give rise to bounded accelerations thereby ensuring good asymptotic behavior for the skew part of the fundamental tensor. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3446-3464 
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    Notes: It is shown that the first-order (Palatini) variational principle for a generic nonlinear metric-affine Lagrangian depending on the (symmetrized) Ricci square invariant leads to an almost-product Einstein structure or to an almost-complex anti-Hermitian Einstein structure on a manifold. It is proved that a real anti-Hermitian metric on a complex manifold satisfies the Kähler condition on the same manifold treated as a real manifold if and only if the metric is the real part of a holomorphic metric. A characterization of anti-Kähler Einstein manifolds and almost-product Einstein manifolds is obtained. Examples of such manifolds are considered. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3465-3475 
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    Notes: We obtain a class of exact solutions representing null particles moving in three-dimensional (anti-) de Sitter spaces by boosting the corresponding static point source solutions given by Deser and Jackiw. In de Sitter space the resulting solution describes two null particles moving on the (circular) cosmological horizon, while in anti-de Sitter space it describes a single null particle propagating from one side of the universe to the other. We also boost the Bañados–Teitelboim–Zanelli black hole solution to the ultrarelativistic limit and obtain the solution for a spinning null particle moving in anti-de Sitter space. We find that the ultrarelativistic geometry of the black hole is exactly the same as that resulting from boosting the Deser–Jackiw solution when the angular momentum of the hole vanishes. A general class of solutions is also obtained which represents several null particles propagating in the Deser–Jackiw background. The differences between the three-dimensional and four-dimensional cases are also discussed. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3476-3489 
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    Notes: Building on the results of previous work [M. A. Clayton, "Canonical general relativity: Diffeomorphism constraints and spatial frame transformations," J. Math. Phys. 39, 3805–3816 (1998)], we demonstrate how matter fields are incorporated into the general linear frame approach to general relativity. When considering the Maxwell one-form field, we find that the system that leads naturally to canonical vierbein general relativity has the extrinsic curvature of the Cauchy surface represented by gravitational as well as nongravitational degrees of freedom. Nevertheless the metric compatibility conditions are undisturbed, and this apparent derivative-coupling is seen to be an effect of working with (possibly orthonormal) linear frames. The formalism is adapted to consider a Dirac Fermion, where we find that a milder form of this apparent derivative-coupling appears. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3490-3498 
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    Notes: All of the homogeneous Lorentzian structures on the oscillator group equipped with a bi-invariant Lorentzian metric, and then the associated reductive pairs, are obtained. Some of them are solutions of the Einstein–Yang–Mills equations. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3499-3508 
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    Notes: A study of radial conformal Killing fields (RCKF) in Minkowski space–time is carried out, which leads to their classification into three disjointed classes. Their integral curves are straight or hyperbolic lines admitting orthogonal surfaces of constant curvature, whose sign is related to the causal character of the field. Otherwise, the kinematic properties of the timelike RCKF are given and their applications in kinematic cosmology is discussed. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3540-3552 
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    Notes: We extend Witten's proof on positive mass conjecture to five-dimensional Lorentzian manifolds with a rigorous treatment in mathematics. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3588-3603 
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    Notes: We give a complete classification of bicovariant first order differential calculi on the quantum enveloping algebra Uq(b+) which we view as the quantum function algebra Cq(B+). Here, b+ is the Borel subalgebra of sl2. We do the same in the classical limit q→1 and obtain a one-to-one correspondence in the finite dimensional case. It turns out that the classification is essentially given by finite subsets of the positive integers. We proceed to investigate the classical limit from the dual point of view, i.e., with "function algebra" U(b+) and "enveloping algebra" C(B+). In this case there are many more differential calculi than coming from the q-deformed setting. As an application, we give the natural intrinsic four-dimensional calculus of κ-Minkowski space and the associated formal integral. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3616-3631 
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    Notes: 4×4 Dirac (gamma) matrices [irreducible matrix representations of the Clifford algebras C(3,1), C(1,3), C(4,0)] are an essential part of many calculations in quantum physics. Although the final physical results do not depend on the applied representation of the Dirac matrices (e.g., due to the invariance of traces of products of Dirac matrices), the appropriate choice of the representation used may facilitate the analysis. The present paper introduces a particularly symmetric real representation of 4×4 Dirac matrices (Majorana representation) which may prove useful in the future. As a by-product, a compact formula for (transformed) Pauli matrices is found. The consideration is based on the role played by isoclinic 2-planes in the geometry of the real Clifford algebra C(3,0) which provide an invariant geometric frame for it. It can be generalized to larger Clifford algebras. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3632-3642 
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    Notes: It is shown that a N×N real symmetric [complex Hermitian] positive definite matrix V is congruent to a diagonal matrix modulo a pseudo-orthogonal [pseudo-unitary] matrix in SO(m,n)[SU(m,n)], for any choice of partition N=m+n. It is further shown that the method of proof in this context can easily be adapted to obtain a rather simple proof of Williamson's theorem which states that if N is even then V is congruent also to a diagonal matrix modulo a symplectic matrix in Sp(N,R)[Sp(N,C)]. Applications of these results considered include a generalization of the Schweinler–Wigner method of "orthogonalization based on an extremum principle" to construct pseudo-orthogonal and symplectic bases from a given set of linearly independent vectors. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3654-3684 
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    Notes: Following the sine–cosine function, the sawtooth wave, square wave, triangular wave, trapezoidal wave, and so on become new easily generated periodic functions in modern electronics. Similar to Fourier's idea, a natural question is whether a signal can be considered as a superposition of easily generated functions with different frequencies. Therefore it is necessary to generalize Fourier analysis based on sine–cosine functions into frequency analysis based on general periodic functions. In this paper, we introduce the frequency series and frequency transformation based on general periodic functions. We discuss when a frequency system is a complete system or an unconditional basis in L2[−π,π], and when a frequency transformation can be carried out in L2(−∞,+∞). For practical convenience almost all easily generated functions in electronics are considered carefully as examples. As a new and practical generalization of classical Fourier analysis, these results will become a theoretical foundation for the technique of easily generated function analysis in signal processing. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3685-3690 
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    Notes: Two remarks on an inverse operator of a differential operator ∂x and on symmetries of two kinds of differential equations Δ(u)=0 and ∂xΔ(u)=0 are pointed out and then a few of remarks about the paper "Generalized W∞ symmetry algebra of the conditionally integrable nonlinear evolution equation" [Y. Lou and J. P. Weng, J. Math. Phys. 36, 3492 (1995)] are presented. It is in particular shown that if we consider ∂x−1 to be a linear and right inverse operator of ∂x, the vector fields σnu(f ) proposed in the above paper are not certain to be symmetries of the Jimbo–Miwa–Kadomtsev–Petviashvili system under consideration. Moreover the commutators of a generalized W∞ algebra established among these vector fields do not always hold. Therefore the vector fields σnu(f ) do not provide an example of application of the formal series ansatz in the above paper. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3691-3692 
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    Notes: To verify the correctness of the high order symmetries for the Jimbo–Miwa–Kadomtsev–Petviashvilli system (JMKP) and the potential JMKP (PJMKP) system given in our paper [J. Math. Phys. 36, 3492–3497 (1995)], Ma's definition of ∂x−1 may be used only for the general nonkernel solutions of the models. For some types of special solutions which are the kernels of some differential operators, one has to use ∂x−1 as the indefinite operator and selected the integral functions appropriately.© 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3693-3696 
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    Notes: This note is a comment on the mathematical aspects of the paper by Mostafazadeh [J. Math. Phys. 37, 1218 (1996)]. Its aim is to remove some of the ambiguities and mistakes of the paper by making more transparent the mathematics involved in the calculation of topological charges. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 3697-3697 
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    Notes: © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 2539-2548 
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    Notes: Various aspects of the Nahm equations in three and seven dimensions are investigated. The residues of the variables at simple poles in the seven-dimensional case form an algebra. A large class of matrix representations of this algebra is constructed. The large N limit of these equations is taken by replacing the commutators by Moyal brackets, and a set of nontrivial solutions in a generalized form of Wigner distribution functions is obtained. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 2549-2559 
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    Notes: Adapting ideas of Daubechies and Klauder [J. Math. Phys. 26, 2239 (1985)] we derive a rigorous continuum path-integral formula for the semigroup generated by a spin Hamiltonian. More precisely, we use spin coherent vectors parametrized by complex numbers to relate the coherent representation of this semigroup to a suitable Schrödinger semigroup on the Hilbert space L2(R2) of Lebesgue square-integrable functions on the Euclidean plane R2. The path-integral formula emerges from the standard Feynman–Kac–Itô formula for the Schrödinger semigroup in the ultradiffusive limit of the underlying Brownian bridge on R2. In a similar vein, a path-integral formula can be constructed for the coherent representation of the unitary time evolution generated by the spin Hamiltonian. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 2560-2567 
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    Notes: Using a discrete curve mapping procedure, we reformulate the problem of nonlinear spin dynamics in three different discrete Heisenberg ferromagnetic spin chain models with different magnetic interactions. The procedure helps to rewrite the Landau–Lifshitz equations that govern the dynamics of spins in these ferromagnetic spin systems as equivalent to the integrable discretization of the completely integrable nonlinear Schrödinger family of equations. The elementary spin excitations in these spin systems are governed by lattice solitons. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 2611-2626 
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    Notes: We develop a cannonical quantization for massive vector fields on a globally hyperbolic Lorentzian manifold. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 2584-2610 
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    Notes: The expectation values of Wilson loop products for the pure Euclidean Yang–Mills theory on R×R given by Ashtekar et al. in the article "SU(N) Quantum Yang–Mills Theory in Two Dimensions: A Complete Solution" [J. Math. Phys. 38, 5453 (1997)] are determined directly for all piecewise analytic loops. For that purpose we enlarge their calculations from quadratic lattices to general floating lattices introducing a new kind of loop independence and slightly modifying the regularization scheme. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 2568-2583 
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    Notes: We examine the geometry of the state space of a relativistic quantum field. The mathematical tools used involve complex algebraic geometry and Hilbert space theory. We consider the Kähler geometry of the state space of any quantum field theory based on a linear classical field equation. The state space is viewed as an infinite-dimensional complex projective space. In the case of boson fields, a special role is played by the coherent states, the totality of which constitutes a nonlinear submanifold C of the projective Fock space PF. We derive the metric on C induced from the ambient Fubini–Study metric on PF. Arguments from differential geometric, algebraic, and Kählerian points of view are presented, leading to the result that the induced metric is flat, and that the intrinsic geometry of C is Euclidean. The coordinates for the single-particle Hilbert space of solutions are shown to be complex Euclidean coordinates for C. A transversal intersection property of complex projective lines in PF with C is derived, and it is shown that the intrinsic geodesic distance between any two coherent states is strictly greater than the corresponding geodesic distance in the ambient Fubini–Study geometry. The functional metric norm of a difference field is shown to give the intrinsic geodesic distance between two coherent states, and the metric overlap expression is shown to measure the angle subtended by two coherent states at the vacuum, which acts as a preferred origin in the Euclidean geometry of C. Using the flatness of C we demonstrate the relationship between the manifold complex structure on PF and the quantum complex structure viewed as an active transformation on the single-particle Hilbert space. These properties of C hold independently of the specific details of the single-particle Hilbert space. We show how C arises as the affine part of its compactification obtained by setting the vacuum part of the state vector to zero. We discuss the relationship between unitary orbits and geodesics on C and on PF. We show that for a Fock space in which the expectation of the total number operator is bounded above, the coherent state submanifold is Kähler and has finite conformal curvature. © 1999 American Institute of Physics.
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    Journal of Mathematical Physics 40 (1999), S. 2627-2639 
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    Notes: For specific choice of parameters the spectrum of the discrete quantum pendulum-integral contains the eigenvalues of a finite matrix which depends analytically on the flux. Under natural continuity assumptions these eigenvalues include the spectral values which may be obtained by the algebraic Bethe ansatz. © 1999 American Institute of Physics.
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